# Nonexamples Of Coarsely Bounded Generation

29 Aug 2022

This is the sixth post in a series on Katie Mann and Kasra Rafi’s paper Large-scale geometry of big mapping class groups. The purpose of this post is to discuss when a Polish group is generated by a coarsely bounded set, and give examples of mapping class groups which are locally coarsely bounded but fail this criterion.

## The general case

We have the following theorem of Rosendal.

Theorem 1.2 (Rosendal). Let $$G$$ be a Polish group. Then $$G$$ is generated by a coarsely bounded set if and only if $$G$$ is locally coarsely bounded and not the countably infinite union of a chain of proper, open subgroups.

Let’s make sense of this. Note that countable, discrete groups are Polish. Being generated by a coarsely bounded set is the analogue of finite generation for discrete groups, so we should expect that for a countable discrete group (which is automatically locally coarsely bounded, having finite discrete neighborhoods of the identity) being finitely generated is equivalent to not being the countably infinite union of a chain of proper subgroups, which are automatically open.

Indeed, suppose $$G$$ is finitely generated by a set $$S$$, and that $$G_1 < G_2 < G_3 < \cdots$$ is a strictly increasing chain of proper subgroups. Since the union of the $$G_i$$ is $$G$$, each $$s \in S$$ belongs to some $$G_i$$, but then at some finite stage $$G_n$$, we have each $$s \in S$$ contained in $$G_n$$, so actually $$G_n = G$$.

Conversely, supposing every strictly increasing chain of proper subgroups terminates, we show that $$G$$ is finitely generated. Indeed, we can take a sequence of finitely generated subgroups! Begin with $$G_1 = \langle s_1 \rangle$$ for some $$s_1 \in G$$. At each stage, add $$s_{n+1} \notin G_n$$ and take $$G_{n+1} = \langle s_1,\ldots,s_{n+1}\rangle$$. Since this sequence terminates, we’ve proven that $$G$$ is finitely generated. (This proof contains the useful fact that every generating set for a finitely generated group contains a \emph{finite} generating set.)

## A non-example: limit type

Remember the “Great Wave off Kanagawa” surface from the previous post? It had genus zero and end space homeomorphic to $$\omega^\omega + 1$$ in the order topology. Take the connect sum of two copies of that surface; so the genus-zero surface with end space homeomorphic to $$\omega^\omega \cdot 2 + 1$$. It looks a little like this: The original “Great Wave” surface has self-similar end space and genus zero, so has coarsely bounded mapping class group. This surface $$\Sigma$$ has locally coarsely bounded mapping class group, but we will show that $$\operatorname{Map}(\Sigma)$$ is not generated by a coarsely bounded set. Consider the index-two subgroup $$G$$ of $$\operatorname{Map}(\Sigma)$$ comprising those mapping classes that fix pointwise the two maximal ends. We’ll show that $$G$$ is a countable union of proper open subgroups $$G_0 < G_1 < \cdots$$. Since $$G$$ has index two in $$\operatorname{Map}(\Sigma)$$, this will show that $$\operatorname{Map}(\Sigma)$$ is also a countable union of proper open subgroups $$G'_0 < G'_1 < \cdots$$, where each $$G'_i$$ is obtained from $$G_i$$ by adding a fixed mapping class $$\phi$$ that swaps the two maximal ends of $$\Sigma$$. (Recall that a subgroup is open if and only if it contains an open neighborhood of the identity, so the $$G'_i$$ are open since $$G$$ is open in $$\operatorname{Map}(\Sigma)$$ and the $$G'_i$$ are open in $$G$$.)

To start, consider a simple closed curve $$\alpha$$ separating $$\Sigma$$ into two pieces, each containing exactly one maximal end. The identity neighborhood we consider is $$U_A$$, where $$A$$ is an annular subsurface with core curve $$\alpha$$. Since $$U_A$$ is a subgroup of $$G$$, we’ll let $$G_0 = U_A$$. Since $$\operatorname{Map}(\Sigma)$$ and hence $$G$$ is Polish, there is a countable dense subset $$\{\phi_i : i \in \mathbb{N}\}$$ of $$G$$. Any open subgroup containing the $$\phi_i$$ is in fact all of $$G$$, so consider the sequence of open subgroups $$G_1 \le G_2 \le \cdots$$, where

$G_i = \langle G_0, \phi_1,\ldots, \phi_i \rangle.$

If we can show that each $$G_i$$ is a proper subgroup of $$G$$, we will be done, even though a priori this chain may not be strictly increasing. Consider a maximal end $$\xi$$ and a neighborhood basis of $$\xi$$ comprising nested clopen neighborhoods $$U_j$$ with $$U_{j+1} \subset U_j$$, beginning with $$U_0$$ being the end set of the component of $$\Sigma - A$$ containing $$\xi$$. In the figure we can think of the clopen neighborhoods as coming from the “fluting” process described in the previous post. Thus, $$U_0 - U_j$$ contains points homeomorphic to $$\omega^{j-1} + 1$$ but not points homeomorphic to $$\omega^j + 1$$. In plainer words, $$U_0 - U_1$$ contains isolated planar ends, $$U_0 - U_2$$ contains ends accumulated by isolated planar ends, $$U_0 - U_3$$ contains ends accumulated by ends accumulated by isolated planar ends, so on and so forth. Anyway, consider $$\phi_1,\ldots,\phi_n$$. Since each $$\phi_i$$ leaves $$\xi$$ invariant, we claim that there exists $$M$$ large such that for all $$m \ge M$$, ends homeomorphic to $$\omega^m + 1$$ contained in $$U_m$$ actually remain inside $$U_m$$ under each $$\phi_i$$. To see this, note that if there was a sequence of ends $$\{\xi_m\}$$ with each $$\xi_m$$ homeomorphic to $$\omega^m + 1$$ such that each $$\xi_m$$ was moved outside of $$U_m$$ by some $$\phi_i$$, then since the sequence $$\{\xi_m\}$$ necessarily converges to $$\xi$$, by the pigeonhole principle some $$\phi_i$$ would have to move $$\xi$$. This already shows us that $$G_i$$ is a proper subgroup of $$G$$, since by the classification of surfaces we can move some $$\xi_m$$ with $$m > M$$ outside of $$U_m$$ (and into a neighborhood of the other maximal end).

This surface $$\Sigma$$ is an example of the general phenomenon Mann–Rafi term having end space of “limit type”. The argument we just gave generalizes to show that if $$\Sigma$$ has limit type (see Definition 6.2 of their paper for a precise definition) then $$\operatorname{Map}(\Sigma)$$ cannot be generated by a coarsely bounded set.

## A non-example: infinite rank

If $$G$$ is a finitely generated group, notice that all (a fortiori continuous) quotients of $$G$$ are finitely generated, and that conversely if $$G$$ has a quotient that is not finitely generated, then $$G$$ cannot be finitely generated. The same is true of Polish groups and coarsely bounded generation: if $$G$$ is a Polish group that has a continuous quotient which is not coarsely boundedly generated, then $$G$$ is not either. A prime example of such a group as a quotient is the countably infinite group $$\bigoplus_{n = 1}^\infty \mathbb{Z}$$.

It is possible to build continuous maps to $$\bigoplus_{n=1}^\infty \mathbb{Z}$$ by using the topology of the end space. Here is one example. Consider the ends $$\xi_n$$ constructed in the previous post which are pairwise noncomparable. The $$\xi_n$$ are maximal ends of self-similar surfaces, so have stable neighborhoods. We form a surface by “fluting” together the union of countably infinitely many copies of each $$\xi_n$$. Necessarily each collection of ends locally homeomorphic to $$\xi_n$$ converges to the maximal end of the flute. Now, this surface is self-similar, with genus zero or infinity, hence has coarsely bounded mapping class group. So take the connect sum of two copies of this surface, and call the connect sum $$\Sigma$$.

As before, take a simple closed curve $$\alpha$$ that separates $$\Sigma$$ into two pieces, each one containing a single one of the two maximal ends. Pick one of the maximal ends, call it $$\xi$$, and let $$U$$ be the neighborhood of $$\xi$$ determined by $$\alpha$$. We claim that for each end $$\xi_n$$ and any mapping class $$\phi$$ belonging to the index-two subgroup of $$\operatorname{Map}(\Sigma)$$ fixing $$\xi$$, the number of ends locally homeomorphic to $$\xi_n$$ mapped into and out of $$U$$ is finite. Indeed, were either quantity infinite, the same argument as in the previous example shows that $$\phi$$ would have to move $$\xi$$.

Anyway, count up the number of ends of type $$\xi_n$$ moved into $$U$$ by $$\phi$$ and subtract the number of ends of type $$\xi_n$$ moved out of $$U$$ by $$\phi$$. This defines a homomorphism $$\ell_n \colon G \to \mathbb{Z}$$. By “shifting a strip of ends locally homeomorphic to $$\xi_n$$”, we can show that $$\bigoplus_{n=1}^\infty \ell_n\colon G \to \bigoplus_{n=1}^\infty \mathbb{Z}$$ is surjective, and continuous, since if $$A$$ is an annular subsurface with core curve $$\alpha$$, the open set $$U_A$$ is contained in the kernel of $$\bigoplus_{n=1}^\infty \ell_n$$.

This surface $$\Sigma$$ is an example of the general phenomenon Mann–Rafi term having end space of “infinite rank”. The argument we just gave generalizes to show that if $$\Sigma$$ has infinite rank (see Definition 6.5 for a precise definition) then $$\operatorname{Map}(\Sigma)$$ cannot be generated by a coarsely bounded set.